Imposing only boundedness constraints on \(P\) , we study excess risk bounds of least squares performed over the class \(\mathcal {W}_{b}\) , denoted in what follows by \(\widehat{\omega }_{b}^{\operatorname{ERM}}\) . A baseline for our work is a conjecture proposed in [1]} postulating the statistical optimality of the constrained least squares estimator \(\widehat{\omega }^{\operatorname{ERM}}_{b}\)  in a sense that it matches the lower bound (REF ). For some of the recent discussions and attempts to resolve this conjecture see, for example, the works [2]}, [3]}, [4]}, [5]}.
The existing results, however, only partially address this conjecture,
restricting to the regimes where \(br \sim m\)  (the notation \(a \sim b\)  means \(a \lesssim b \lesssim a\) ). Specifically, the best known guarantees that can be obtained, for instance, via localized Rademacher complexity arguments [6]}, [7]}, [8]} yielding the following upper bound
\(\operatorname{\mathbb {E}}R(\widehat{\omega }^{\operatorname{ERM}}_{b}) - \inf \limits _{\omega \in \mathcal {W}_{b}}R(\omega )\lesssim \frac{dm^2}{n} + d\cdot \frac{r^2b^2}{n},\) 
